Separate phase space into areas of increasing and decreasing prey or predator abundance.
March 21, 2022
Isoclines …
Lotka-Volterra
If isoclines are perpendicular … eternal oscillations.
Lotka-Volterra with Carrying Capacity
If isoclines tilts … damped oscillations
What if prey isocline rises and falls?
On the left … Allee effect and Prey switching
On the right … Density dependence
In the middle - stable oscillations
In the right - damped oscillations
In the left - unstable oscillations
So …. if you increase carrying capacity
You can send the system into an unstable state and INCREASE the probability of extinction.
Rosenzweig’s “enrichment paradox”
“…increasing the supply of limiting nutrients or energy tends to destroy the steady state. Thus man must be very careful in attempting to enrich an ecosystem in order to increase its food yield. There is a real chance that such activity may result in decimation of the food species that are wanted in greater abundance.” - Rosenzweig (1971)
Nice theory you’ve got there ….
But is it real?
More experiments with Didinium nausutum vs. Paramecium caudatum.
High carrying capacity
Luckinbill 1973 recreates Gause experiments with a different food source.
Low carrying capacity
When the food source is limited … some nice oscillations!
Consistent with Enrichment Paradox …
Harrison (1995) reinterprets the data
x`
HOWEVER, in natural systems
efforts to show this effect generally fail. (reviewed in Roy and Chattopadhyay 2007).
Because things are complicated! Fluctuations are buffered by:
- Anti-predation strategies
- Multple prey species
- Ratio-dependent responses
- Refuge effect
- Predation capacity
Stabilizing effect of refuge, graphically
As long as there is some predator free areas, the oscillations are stabilized.
Refuge effect
Example: Wildlife management “stumbled” in to this with wildlife reserves, the National Wildlife Refuge System, etc. On the one hand, people actively manage their land to attract game / fowl enrichment, on the other hand there are pockets where populations can always recover.
Stabilizing effect of predator carrying capacity
Embracing complexity
Lots of fundamental, heavily cited papers on the relative roles of predation, competition and external environmental effects on intertidal rocky communities.
Later (2008-2016) secretary of National Ocean and Atmosphere Administration (NOAA).
Introducing Functional Response…
A functional response is the intake rate of a consumer as a function of food density.
A more realistic functional response
\[f(V) = {aV \over 1+ahV}\]
Where \(a\) is attack rate, \(h\) is handling time.
This assumes a single predator species focusing on a single prey species, getting slowed down by processing time.
Holling’s “disk” equation
An assistant picked sandpaper discs off a tackboard (Holling 1951).
Fundamental foundations for predation theory ensued.
Another functional response
Here, we assume that when prey are very scarce, there are other options for predators, or non-linear reasons why they’re harder to find.
Number of predators to prey abundance
Prey Response to Predation
Total Numerical Prey Response = Prey eaten / predator X predator response
Total Proportional Prey Response = Total prey response / Prey abundance
Prey deaths
This curve will (typically) peaks at some intermediate number, where:
- (
1) - N. Predators is (already) high, - (
2) - N. of Prey consumed per Predator is high (approaching asymptote),
but
- (
3) - Total N of prey is NOT too high.
Prey growth
All of that is total mortality of prey. Now - what we need is proporional growth of prey. We have a good, time-tested model for that:
Combining Growth with Removal
Where the two lines intersect, Growth = Removal , we’re at equilibrium. The question is … are these equilibria stable?
Stability of equilibria
Follow the arrows!
As usual, super elegant theory … but what is it good for?
What limits moose?
Big Q: Are herbivore populations controlled bottom-up (food limited) or top-down (predator limited).
Models from theory
Food regulates
Models from theory
Models from theory
Models from theory
Meta-analysis
Data aggregated from many sites with high and low moose and wolf densities.
Empirically estimated …
Functional response curve
Looks Type II-ish, which makes sense since moose is (often) the main prey of wolves, and wolves are (often) the main predator of moose.
Density dependent growth curve
Note: these data are only from Isle Royale (Lake Superior) long term study.
Put it all together …
low productive environment (3) system more likely to be predator limited
more productive environment (1) more likely to be food-limited
a 2-state model is unlikely ….
… but the numeric response curve is shallow. Meaning a local system (when perturbed) can switch from one to the other.
Consistent with observed variability not just in space but in time.